L(s) = 1 | + 3-s − 4·5-s + 9-s + 2·11-s + 6·13-s − 4·15-s + 4·17-s + 4·19-s + 2·23-s + 11·25-s + 27-s − 2·29-s + 2·33-s + 2·37-s + 6·39-s − 4·43-s − 4·45-s − 12·47-s + 4·51-s − 6·53-s − 8·55-s + 4·57-s + 8·59-s − 6·61-s − 24·65-s − 8·67-s + 2·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.78·5-s + 1/3·9-s + 0.603·11-s + 1.66·13-s − 1.03·15-s + 0.970·17-s + 0.917·19-s + 0.417·23-s + 11/5·25-s + 0.192·27-s − 0.371·29-s + 0.348·33-s + 0.328·37-s + 0.960·39-s − 0.609·43-s − 0.596·45-s − 1.75·47-s + 0.560·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-s + 1.04·59-s − 0.768·61-s − 2.97·65-s − 0.977·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.486002552\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.486002552\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.95477629177032576079496808423, −9.694844863101715383876080158734, −8.708229597668417567477010531317, −8.077019225803717160884657215512, −7.38098041713554163025681861500, −6.34426698354997241941479748412, −4.87924854330324804890540113594, −3.65675458100353588864150742280, −3.37164547236104444198213794231, −1.14653761218391715748516515519,
1.14653761218391715748516515519, 3.37164547236104444198213794231, 3.65675458100353588864150742280, 4.87924854330324804890540113594, 6.34426698354997241941479748412, 7.38098041713554163025681861500, 8.077019225803717160884657215512, 8.708229597668417567477010531317, 9.694844863101715383876080158734, 10.95477629177032576079496808423